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Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory

Keita Allen, Lucas Piessevaux

TL;DR

This work extends the nonequivariant synthetic reconstruction program to finite-abelian-group-equivariant motivic homotopy theory over C. After p-completion, the cellular A-equivariant motivic category SH^A(C)^cell is equivalent to a topological category of synthetic A-spectra Syn^A, with the deformation controlled by a τ-parameter that encodes equivariant Adams–Novikov data. A central theme is the construction of equivariant algebraic cobordism MGL_A via global group laws, its regularity, and a Chow-structure that yields a clean heart identified with comodule categories over MU-cooperations; this underpins both a pure reconstruction and a Betti-realisation bridge to non-motivic data. The authors connect SH^A(C)^cell to Syn^A through a Betti-realisation functor Be^A, show Be^A has covering lifting properties, and prove a robust equivariant perfect even filtration whose module category recovers Syn^A as modules over a filtered Sp^A. Collectively, the results provide a coherent, purely topological framework for understanding equivariant motivic phenomena, encode the equivariant Adams–Novikov descent in the synthetic setting, and yield a controlled, modular reconstruction that integrates Chow-theoretic and formal-group-law techniques with isotropy separation and Betti realization.

Abstract

We prove a topological reconstruction result for the category of cellular $A$-equivariant motivic spectra over the complex numbers where $A$ is a finite abelian group: after completion at an arbitrary prime, this is equivalent to the completion of a category of synthetic $A$-equivariant spectra. The latter is a deformation of equivariant spectra which categorifies the equivariant perfect even filtration and is closely related to the equivariant Adams--Novikov spectral sequence. Our main computational input is a description of the bigraded homotopy groups of equivariant algebraic cobordism in terms of equivariant formal group laws.

Synthetic equivariant spectra for finite abelian groups and motivic homotopy theory

TL;DR

This work extends the nonequivariant synthetic reconstruction program to finite-abelian-group-equivariant motivic homotopy theory over C. After p-completion, the cellular A-equivariant motivic category SH^A(C)^cell is equivalent to a topological category of synthetic A-spectra Syn^A, with the deformation controlled by a τ-parameter that encodes equivariant Adams–Novikov data. A central theme is the construction of equivariant algebraic cobordism MGL_A via global group laws, its regularity, and a Chow-structure that yields a clean heart identified with comodule categories over MU-cooperations; this underpins both a pure reconstruction and a Betti-realisation bridge to non-motivic data. The authors connect SH^A(C)^cell to Syn^A through a Betti-realisation functor Be^A, show Be^A has covering lifting properties, and prove a robust equivariant perfect even filtration whose module category recovers Syn^A as modules over a filtered Sp^A. Collectively, the results provide a coherent, purely topological framework for understanding equivariant motivic phenomena, encode the equivariant Adams–Novikov descent in the synthetic setting, and yield a controlled, modular reconstruction that integrates Chow-theoretic and formal-group-law techniques with isotropy separation and Betti realization.

Abstract

We prove a topological reconstruction result for the category of cellular -equivariant motivic spectra over the complex numbers where is a finite abelian group: after completion at an arbitrary prime, this is equivalent to the completion of a category of synthetic -equivariant spectra. The latter is a deformation of equivariant spectra which categorifies the equivariant perfect even filtration and is closely related to the equivariant Adams--Novikov spectral sequence. Our main computational input is a description of the bigraded homotopy groups of equivariant algebraic cobordism in terms of equivariant formal group laws.
Paper Structure (51 sections, 115 theorems, 249 equations)

This paper contains 51 sections, 115 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Global methods and a motivic refinement
  3. An equivariant Chow t-structure
  4. Relation to the perfect even filtration
  5. Equivariant motivic homotopy theory
  6. Recollections
  7. Isotropy separation
  8. Absolute motivic spectra
  9. Equivariant algebraic cobordism
  10. Motivic Thom spectra
  11. An absolute refinement
  12. The equivariant cell structure
  13. Incomplete algebraic cobordism spectra
  14. Algebraic cobordism and global group laws
  15. Global group laws
  16. ...and 36 more sections

Key Result

Theorem 1

Let $p$ be an arbitrary prime, then there is an equivalence where $\mathrm{Syn}$ is the category of synthetic spectra.

Theorems & Definitions (313)

  • Theorem : gheorghe2022c at $p = 2$, pstragowski_synthetic_2022 at all primes
  • Theorem A: \ref{['thm: synthetic reconstruction']}, \ref{['thm: generic fiber']}, \ref{['cor: modules over cofiber of tau']}
  • Theorem : hausmann2022global
  • Proposition B: \ref{['prop: MGL is absolute']}
  • Theorem C: \ref{['prop: ggl on MGL']}, \ref{['cor: homotopy groups of gfp of mgl']}, \ref{["cor : universal map of ggl's"]}, \ref{['prop: vanishing in MGL']}, \ref{['prop: identification of the Chow line']}, \ref{['prop: mod p^i equivariant HKO']}
  • Theorem D: \ref{['prop: Chow weight structure']}
  • Theorem E: \ref{['thm: Chow heart structure']}
  • Theorem F: \ref{['cons: Chow t-structure']}, \ref{['cor: identification of the heart']}, \ref{['cor: identification of SF']}
  • Theorem G: \ref{['cor: clp']}, \ref{['lem: common envelope']}, \ref{['thm: pure reconstruction']}
  • Theorem H: \ref{['prop: filtered model']}, \ref{['thm: identification of even filtration on unit']}
  • ...and 303 more